1.4 Financial applications of geometric sequences questions for IB Maths AA

Write down the value of $x$.

Expand $(1-x{)}^5$.

Use the first four terms of your expansion to estimate the value of the car after $5$ years. Give your answer correct to the nearest dollar.

Find the value of $p$, the value of $q$ and the value of $r$.

Use the table to write down an estimate of ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }A(n)}$, correct to two decimal places.

A second account pays a nominal annual interest rate of $5.13\mathrm{\%}$, compounded once per year. Kofi claims that if the first account compounded interest often enough, it would give him more money after one year than the second account.

Use your answer to part (b) to determine whether Kofi is correct.

Use your GDC to find the values of $g(0.1)$, $g(0.01)$ and $g(0.001)$, giving your answers correct to five significant figures.

Use l'Hôpital's rule to show that ${\displaystyle \underset{x\to 0}{\lim }\frac{\ln (1+x)}{x}=1}$.

Hence show that ${\displaystyle \underset{x\to 0^{+}}{\lim }g(x)=e}$.

Let $r$ be a positive constant. Show that ${\displaystyle \underset{x\to 0^{+}}{\lim }(1+rx{)}^{\frac{1}{x}}=e^r}$.

Ana invests $1000 for one year at a nominal annual interest rate of $6\mathrm{\%}$, compounded $n$ times per year. The value of her investment at the end of the year is $1000{(1+{\displaystyle \frac{0.06}{n}})}^n$ dollars.

Use part (d) to find the limit of the value of her investment as $n\to \mathrm{\infty }$. Give your answer correct to two decimal places.

Let $E(x)=e-g(x)$, for $x>0$.

Use your GDC to find the values of ${\displaystyle \frac{E(x)}{x}}$ for $x=0.1$, $x=0.01$ and $x=0.001$, giving your answers correct to four significant figures.

Suggest an exact value for ${\displaystyle \underset{x\to 0^{+}}{\lim }\frac{E(x)}{x}}$.

Use the Maclaurin series for $\ln (1+x)$ to show that ${\displaystyle \frac{\ln (1+x)}{x}}=1-{\displaystyle \frac{x}{2}}+{\displaystyle \frac{x^2}{3}}-\dots$

Hence show that $g(x)=e(1-{\displaystyle \frac{x}{2}}+{\displaystyle \frac{11x^2}{24}}+\dots )$.

Hence prove that your answer to part (g) is correct.

Use the approximation $E(x)\approx {\displaystyle \frac{ex}{2}}$ with $x={\displaystyle \frac{1}{n}}$ to estimate the least value of $n\in {\mathbb{Z}}^{+}$ for which $e-{(1+{\displaystyle \frac{1}{n}})}^n<0.001$.

Use your GDC to find the least value of $n\in {\mathbb{Z}}^{+}$ for which $e-{(1+{\displaystyle \frac{1}{n}})}^n<0.001$.

Use the series for $g(x)$ from part (i) to explain why the estimate in part (k)(i) is too large.

Find the value of the car after $2$ years.

Show that the amount in Lina's savings account after $3$ years is €6655.

Determine the least number of complete years after which the amount in Lina's savings account is greater than the value of the car.

Find the value of Elena's investment after $7$ years. Give your answer correct to two decimal places.

Find the interest Elena earns during these $7$ years.

Find the number of complete years after which the value of Elena's investment first exceeds €9000.

Write down an expression for the value of each investment after $n$ years.

The two investments have the same value after $n$ years. Find the exact value of $n$, giving your answer in the form $\frac{\ln a}{\ln b}$, where $a,b\in \mathbb{Q}$.